8th ASCE Specialty Conference on Probabilistic Mechanics and Structural Reliability
PMC2000-124
CLOSED-LOOP IDENTIFICATION AND CONTROL RE-DESIGN: AN EXPERIMENTAL STRUCTURAL CONTROL EXAMPLE∗ G. Jin, M. K. Sain, B. F. Spencer, Jr. M. ASCE University of Notre Dame, Notre Dame, IN 46556
[email protected],
[email protected],
[email protected] Abstract Closed-loop identification and control re-design have been investigated intensively recently. However current results are often theoretically oriented and lack concrete application support. In this paper, we provide an efficient solution procedure to this problem in the area of structural control, and apply it to the control design of a bench-scale experimental system. The main content of the paper is divided into three parts: the recovery of system transfer functions from closed-loop data, the state space system identification, and the LQG control re-design. A motivation of this research is to analyze and reduce unexpected shifts of the 2nd and 3rd modal frequencies when the control-loop is closed. The experimental results show that our approach helps to achieve this with improved control performance by reducing the model error from open-loop identification.
Introduction The dual topics of identification for control and model based control design have attracted extensive discussions for the past two decades, which naturally lead to the currently innovative concept of integrated system identification and control design. The idea is to identify the plant model in closed-loop with a feedback controller running, and based on this model (and possibly the closed-loop performance, and the working controller), the controller is re-designed aiming at improved closed-loop performance and/or stability margins. The arguments for this usually iterative procedure lie in the facts that the model estimated in open-loop may not adequately describe the system behavior under closed-loop conditions, e.g. when the system has significant nonlinear characteristics, and that by iterating on the identification-design procedure, the identified model may be optimally shaped for the control algorithm to produce a high-performance controller. Two fundamental issues thus arise: how to identify the plant in closed-loop condition, and how to incorporate the closed-loop results into the design algorithm. The closed-loop identification is a well-recognized problem. A comprehensive review with prediction error method is given in (Forssell and Ljung, 1999); the observer based techniques for flexible structure identification are discussed in (Juang and Phan, 1994) and (Phan et al, 1994). For the issue of control design based on the closed-loop results, several tentative plans exist in literature: the modal cost analysis approach of (Zhu and Skelton, 1994), and the LQG-oriented algorithm (Zang et al, 1995). Van den Hof and Heuberger (1995) gives a nice review of the integrated identificationdesign procedure from a more theoretical point of view. This paper develops, for structural control applications, an efficient solution procedure to the closed-loop problems discussed so far. The research work is motivated by the control design of an experimental structural system, which contains a 3-story building model and ∗
Manuscript submitted March 15, 2000 to the Eighth ASCE Specialty Conference on Probabilistic Mechanics and Structural Reliability. This work was supported in part by the Frank M. Freimann Chair in Electrical Engineering, University of Notre Dame, and by the National Science Foundation under Grants CMS 93-01584, CMS 95-00301, CMS 95-28083 and CMS 99-00234. Jin, Sain, Spencer
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Figure 1. Experiment Setup
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Figure 2. Model FRF: Open-Loop v.s. Closed-Loop
two Active Mass Drivers (AMDs). See Figure 1. The AMDs are installed in the structure to reduce the vibration of the building due to a simulated earthquake input. In a previous study (Jin et al, 2000b), an LQG controller has been designed using the open-loop identified model. The closed-loop response achieves satisfactory vibration reduction, but the resonant frequencies of the 2nd and 3rd modes have been unexpectedly increased compared with the uncontrolled case. (This is shown in Figure 6.) It is observed that the magnitude of the AMD frequency response function (FRF) varies with different excitation levels; and that when installed in the structure, the displacement level of the AMD differs significantly between the open-loop and closed-loop tests. This suggests that the model obtained in the open-loop identification may not precisely predict the system behavior in the closedloop, which makes the closed-loop solution an attracting alternative. The rest of the paper proceeds as follows. In the next section, we will derive the formula to recover the system transfer functions from closed-loop FRF data. Based on this closed-loop estimation, the polynomial curve-fitting technique and the Eigensystem realization algorithm (ERA) will be used to identify a state space model of the system. Finally, the LQG control re-design problem will be discussed in the last section, along with the experimental results. Transfer Function Estimation from Closed-Loop Data Refer to the system closed-loop diagram in Figure 3. The output measurements include the absolute accelerations of the structure y = [¨ x1 , x¨2 , x ¨3 ]0 , and the relative displacement of the AMD (w.r.t. the supporting floor) z = [xa1 , xa2 ]0 . The inputs to the system contain the reference signals to the AMD r = [ua1 , ua2 ]0 , and the earthquake disturbance v = x¨g . To estimate the plant transfer function P , the closed-loop transfer function T (from [r 0 , v 0 ]0 to [y 0, z 0 ]0 ) and sensitivity function S (from [r 0 , v 0 ]0 to u) are measured with a spectrum analyzer. Now let P , T , and S be divided according to the diagram in Figure 3 as "
P = [P1 P2 ] = Jin, Sain, Spencer
P11 P12 P21 P22
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"
, T = [T1 T2 ] =
T11 T12 T21 T22
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, S = [S1 S2 ]
(1) 2
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Figure 3. Closed-Loop Diagram
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Figure 4. Poles: Open- v.s. Closed-Loop
Then we have, "
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Substituting u = S1 r + S2 v into (2), we have "
y z
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P11 S1 P21 S1
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By using the closed-loop expression "
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and the fact that r and v are independent, P is solved by combining (3) and (4), P = [ P1 P2 ] = [ T1 S1−1 T2 − T1 S1−1 S2 ]
(5)
Equation (5) provides the formula to recover P from T and S. Note that in the above derivations, the controller K is assumed to be unknown. Moreover, as shown in (5), it is not necessary to distinguish between the measurements y and z in order to calculate P . The experimental results of estimating P from open-loop data (blue line, or darker line if printed in black-white) and closed-loop data (red line, or lighter line if printed in black-white) are compared in Figure 2. The curves have relatively the same shape, which indicates that the closed-loop estimate is consistent with that of the open-loop. However, the gain differences between the two curves strengthen our initial conjecture that the system behaves differently between open-loop and closed-loop operations. State Space System Identification In this section, we will outline the procedure to identify a state space model for the estimated P in (5). To start, the polynomial matrix parameterization is used to curve-fit the FRF data. This amounts to find an optimal transfer function matrix in the form of G(z −1 ) = Jin, Sain, Spencer
B(z −1 ) α(z −1 )
(6) 3
with B(z −1 ) = B0 +B1 z −1 +B2 z −2 +· · ·+Bp z −p being the numerator polynomial matrix, and α(z −1 ) = 1 + a1 z −1 + a2 z −2 + · · · + ap z −p the denominator polynomial, such that the frequency weighted (by w(ωi )) estimation error is minimized, i.e. G∗ (z −1 ) = arg min G
l X i=1
w(ωi )2 ||P (ωi) − G(z −1 (ωi ))||2F
(7)
Here || · ||F denotes the Frobenius norm. The parameterization (6) is chosen due to the facts that it is easy to incorporate pre-specified zeros of the transfer functions, and that efficient algorithms exist to solve for (7), see (Jin et al, 2000a) and the references therein. Once the curve-fitting stage is completed, the Markov parameters Y0 , Y1 , Y2 , . . . of the system may be derived by the following iterative calculations starting from Y0 = B0 (Bayard 1992): (
Yk =
P
Bk − ki=1 ai Yk−i , for k = 1, . . . , p P − pi=1 ai Yk−i, for k = p + 1, . . . , ∞
(8)
Now the ERA method (Juang 1994) is used to find a state space realization (A, B, C, D) from the Markov parameters. First form the generalized Hankel matrices H(k − 1) =
Yk Yk+1 .. .
Yk+1 Yk+2 .. .
... ... .. .
Yk+β2−1 Yk+β2 .. .
Yk+β1−1 Yk+β1 . . . Yk+β1+β2 −2
(9)
Note that in general, β1 and β2 are chosen to be the smallest numbers such that H(k) has as large row and column ranks as possible. Additional suggestions to determine their optimal values are given in (Juang 1994). Let the singular value decomposition of H(0) be H(0) = UΣV T , and let n denote the desired order of state space realization. Then H(0) can be approximated by H(0) ≈ Un Σn VnT (10) where Un and Vn are the first n columns of U and V respectively, and Σn is the diagonal matrix containing the largest n singular values of H(0). Finally, an n-th order state space realization (A, B, C, D) can be calculated by −1
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T A = Σn 2 UnT H(1)Vn Σn 2 , B = Σn2 VnT Er , C = Em Un Σn2 , D = Y0
(11)
T where Er and Em are the elementary matrices that pick out the first r (the number of system inputs) columns and first m (the number of system outputs) rows of their multiplicands respectively. The above procedure is applied to solve for a discrete time state space model with adjustable parameters chosen as p = 20 in (6), β1 = 24, β2 = 40 in (9), and n = 20 in (10). We also pre-specified four zeros at the zero frequency for the transfer functions from each of the AMD command inputs to the structure accelerations and the other AMD displacement outputs, because these zeros are observed in the analytical model of the system (Jin et al, 2000b). We have chosen a larger state space realization (20 states compared to 10 states in the open-loop model) in order to more accurately fit the FRF data. To compare the open- and the closed-loop models, their poles are plotted in Figure 4, with blue-circles for the open-loop model and red-crosses for the closed-loop model. The magnitude frequency response of the identified model (only 3rd floor acceleration output channel is included, due to page limitations) is compared to the FRF data in Figure 5.
Jin, Sain, Spencer
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Figure 5. Comparison between Identified Model and FRF Data
LQG Control Re-Design The model identified from closed-loop data (11) naturally contains more modes than the open-loop one. Thus it is crucial to properly incorporate the new model information into the control design algorithm. It is shown in this section that a simple and efficient way to do this is by choosing the Hankel singular values (HSVs) of the model as the primary design weight for the LQG method. First the (zero-order hold) equivalent continuous-time model calculated from the identified discrete-time model is transformed into the balanced modal representation (Am , Bm , Cm , Dm ) (Gawronski 1998) with T T T n ] , , . . . , Bm Am = diag(Ami ), Bm = [ Bm1 Cm = [ Cm1 , · · · , Cm n2 ] 2
and,
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Ami =
−ξi ωi ωi −ωi −ξi ωi
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, ||Bmi ||F = ||Cmi ||F
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The HSV γi of the i-th mode can be approximated by γi ≈
||Bmi ||2F ||Cmi||2F = 4ξi ωi 4ξi ωi
(14)
with numerical values given in Table 1. Comparing with Figure 4, the five modes with largest HSVs (denoted by * in the table) coincide with the modes obtained in the open-loop tests. Now, since we have ||Gi||2 ||Gi ||∞ ≈ √ (15) γi ≈ 2 2 ξ i ωi where Gi (s) = Cmi (sI − Ami )−1 Bmi is the transfer function of the i-th mode, important modes may be systematically assigned with higher authority by choosing γi as the weight for the i-th mode. The re-designed LQG controller is tested in the closed-loop, and the control performance is evaluated by the measured FRF data from earthquake input x¨g to the structure absolute accelerations x¨i . The results (red-solid lines) are shown in Figure 6, together with that of the original controller (blue-dashed lines) and the uncontrolled case (green-dotted lines). The re-designed controller achieves further suppression of the 2nd and 3rd structure modes with reduced resonant frequencies when compared to the original design. This concludes that model error is indeed a factor in the observed behavior. Jin, Sain, Spencer
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Figure 6. Open- and Closed-Loop FRF Results with Different Controllers
Mode f (hz) HSV
#1 # 2* #3 # 4* # 5* #6 #7 # 8* # 9* # 10 0.569 0.788 0.816 1.409 1.521 2.346 2.341 2.404 3.825 4.050 0.221 9.209 0.363 0.913 1.659 0.771 0.325 2.966 3.320 0.165
Table 1: Resonant Frequencies and HSVs of the Closed-Loop Identified Modes References Bayard D. (1992), “An Algorithm for State-Space Frequency Domain Identification without Windowing Distortions”, Proc. Conference on Decision and Control, pp. 1707-1712. Forssell U., and Ljung L. (1999), “Closed-Loop Identification Revisited”, IFAC Automatica, Vol 35, No 7, P 1215-1241. Gawronski W. (1998), Dynamics and Control of Structures, Springer. Jin G., Sain M.K., and Spencer Jr. B.F. (2000a), “Frequency Domain Identification with Fixed Zeros: First Generation Seismic-AMD Benchmark”, to appear in Proc. American Control Conference. Jin G., Sain M.K., and Spencer Jr. B.F. (2000b), “A Bench-Scale Experiment for AMD-Building Control Systems”, to appear in Proc. ASCE Engineering Mechanics Conference. Juang J-N (1994), Applied System Identification, Prentice-Hall. Juang J-N, and Phan M. (1994), “Identification of System, Observer, and Controller from Closed-Loop Experimental Data”, AIAA Journal of Guidance, Control, and Dynamics, Vol 17, No 1, P 91-96. Phan M. Juang J-N Horta L.G. and Longman R.W (1994), “System Identification from Closed-Loop Data with Known Output Feedback Dynamics” AIAA Journal of Guidance, Control, and Dynamics Vol 17, No 4, P 661-669. Van den Hof P.M.J., and Heuberger P.S.C. (1995), “Identification and Control–Closed-loop Issues”, IFAC Automatica, Vol 31, No 12, P 1751-1770. Zang Z., Bitmead R.R., and Gevers M. (1995), “Iterative Weighted Least-squares Identification and Weighted LQG Control Design”, IFAC Automatica, Vol 31, No 11, P 1577-1594. Zhu G., and Skelton R. (1994), “Integrated Modeling and Control for the Large Spacecraft Control Laboratory Experiment Facility”, AIAA Journal of Guidance, Control, and Dynamics, Vol 17, No 3, P 442-450.
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